Directional splines and their use for smoothing ejections and fractures of interpolant

A method for constructing a directional cubic spline for a set of points on a plane is formulated and proposed. The spline is compared with the Schoenberg $B$-spline, Akima and Catmull–Rom splines. It is shown that for unequally spaced points, in comparison with the $B$-spline, it gives significantly lower overshoots and is practically free of strong kinks, which are characteristic of Akima splines. The spline does not give loops and oscillations, which are a characteristic drawback of parametric splines, in particular, Hermitian ones, which include the Catmull-Rom spline. A fast method for optimizing the spline guiding coefficient is proposed, the purpose of which is to minimize the discontinuities of the second derivative of the function at its intermediate points. An example of optimization of a directional third-order spline is given. A fourth-order directional spline, which is free of kinks, is also proposed. The method of optimization of the directional spline of the fourth order is formulated, the algorithm of its optimization is stated. The optimization criteria are the spline length and the smallest distance between its global maximum and minimum. It is shown that, in comparison with the Schoenberg spline, the fourth-order directional spline has smaller outliers. A method for automatic blunting of sharp peaks of curves is proposed, which can be applied to all types of splines.